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Markov processes conditioned on their location at large exponential times

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  • Evans, Steven N.
  • Hening, Alexandru

Abstract

Suppose that (Xt)t≥0 is a one-dimensional Brownian motion with negative drift −μ. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like X conditioned to hit 0, after which time it behaves as X killed at the last time X visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift +μ when it is negative, like Brownian motion with negative drift −μ when it is positive, and is killed according to the local time spent at 0.

Suggested Citation

  • Evans, Steven N. & Hening, Alexandru, 2019. "Markov processes conditioned on their location at large exponential times," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1622-1658.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:5:p:1622-1658
    DOI: 10.1016/j.spa.2018.05.013
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    Cited by:

    1. Çetin, Umut & Larsen, Kasper, 2023. "Uniqueness in cauchy problems for diffusive real-valued strict local martingales," LSE Research Online Documents on Economics 118743, London School of Economics and Political Science, LSE Library.

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